differential

may be made smaller than any given positive number by making |Δ⁢x| sufficiently small. If we generally denote by ⟨Δ⁢x⟩ an expression having such a property, we can write

f⁢(x+Δ⁢x)-f⁢(x)Δ⁢x-f′⁢(x)=⟨Δ⁢x⟩.

This allows us to express the increment of the functionf⁢(x+Δ⁢x)-f⁢(x):=Δ⁢f in the form

Δ⁢f=(f′⁢(x)+⟨Δ⁢x⟩)⁢Δ⁢x=f′⁢(x)⁢Δ⁢x+⟨Δ⁢x⟩⁢Δ⁢x.

(1)

This result may be uttered as the

Theorem. If the derivative f′⁢(x) exists, then the increment Δ⁢f of the function corresponding to the increment of the argument from x to x+Δ⁢x may be divided into two essentially different parts:
1∘. One part is proportional to the increment Δ⁢x of the argument, i.e. it equals this increment multiplied by a coefficientf′⁢(x) which is on the increment.
2∘. The ratio of the other part ⟨Δ⁢x⟩⁢Δ⁢x to the increment Δ⁢x of the argument tends to 0 along with Δ⁢x.

By Leibniz, the former part f′⁢(x)⁢Δ⁢x is called the differential, or the differential increment of the function, and denoted by d⁢f⁢(x), briefly d⁢f.

It is easily checked that when one has set the tangent line of the curve at the point (x,f⁢(x)), the differential increment d⁢f⁢(x) geometrically means the increment of the ordinate of the corresponding to the transition from the abscissax to the ascissa x+Δ⁢x.